A new and sharper bound for Legendre expansion of differentiable functions
aa r X i v : . [ m a t h . NA ] M a r A new and sharper bound for Legendre expansion ofdifferentiable functions
Haiyong Wang †‡ March 2, 2018
Abstract
In this paper, we provide a new and sharper bound for the Legendre coefficients ofdifferentiable functions and then derive a new error bound of the truncated Legendreseries in the uniform norm. The key idea of proof relies on integration by partsand a sharp Bernstein-type inequality for the Legendre polynomial. An illustrativeexample is provided to demonstrate the sharpness of our new results.
Keywords:
Legendre coefficient, differentiable functions, sharp bound.
AMS classifications:
Let P n ( x ) be the Legendre polynomial of degree n which is defined by P n ( x ) = 12 n n ! d n dx n (cid:2) ( x − n (cid:3) , n ≥ . (1.1)The set of Legendre polynomials { P ( x ) , P ( x ) , · · · } form a system of polynomials or-thogonal on the interval [ − ,
1] with respect to the weight function ω ( x ) = 1 and Z − P n ( x ) P m ( x ) dx = h n δ mn , (1.2)where δ mn is the Kronecker delta and h n = (cid:18) n + 12 (cid:19) − . (1.3) † School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan430074, P. R. China. E-mail: [email protected] ‡ Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University ofScience and Technology, Wuhan 430074, China. f := [ − , → R is defined by f ( x ) = ∞ X n =0 a n P n ( x ) , (1.4)where the Legendre coefficients are given by a n = h − n Z − f ( x ) P n ( x ) dx. (1.5)The problem of estimating the magnitude of the Legendre coefficients a n is of particularinterest both from the theoretical and numerical point of view. Indeed, it is useful notonly in understanding the rate of convergence of Legendre expansion but useful also inestimating the degree of the Legendre polynomial approximation to f ( x ) within a givenaccuracy.When f ( x ) is analytic in a neighborhood of the interval [ − , f ( x ) is a differentiable function. Wefirst establish a new bound for the Legendre coefficients and the key ingredient here isthat the Legendre polynomial satisfies a sharp Bernstein-type inequality. An illustrativeexample is provided to show that our new result is sharper than the result given in [5,Theorem 2.1]. Furthermore, we then derive a new error bound of Legendre expansion inthe uniform norm. Our main results are stated in the next section. In this section we state an explicit and computable bound for the Legendre coefficientsof differentiable functions. This new bound, as will be shown later, is sharper than theone given in [5, Theorem 2.1]. Before proceeding, we first define the weighted semi-norm k f k := Z − | f ′ ( x ) | (1 − x ) dx. (2.1)The following Bernstein-type inequality of Legendre polynomials will be useful. Lemma 2.1.
For x ∈ [ − , and n ≥ , we have (1 − x ) | P n ( x ) | < r π (cid:18) n + 12 (cid:19) − . (2.2)2 oreover, the above inequality is optimal in the sense that the factor ( n + ) − can notbe improved to ( n + + ǫ ) − for any ǫ > and the constant p /π is best possible.Proof. See [1, 2].Figure 1: The ratio of the term on the left-hand side to the term on the right-and sideof (2.2) for n = 2 (left), n = 6 (middle) and n = 18 (right).To show the sharpness of (2.2), we consider the ratio of the term on the left-hand sideto the term on the right-hand side as a function of x . Numerical results are presentedin Figure 1 for three values of n . It is clear to see that the maximum value of the ratiois very close to one.We are now ready to state our first main result on the bound of Legendre coefficientsfor differentiable functions. Theorem 2.2.
Assume that f, f ′ , . . . , f ( m − are absolutely continuous and the m thderivative f ( m ) ( x ) is of bounded variation. Furthermore, assume that V m = k f ( m ) k < ∞ .Then, for n ≥ m + 1 , | a n | ≤ V m p π (2 n − m − m Y k =1 h n − k . (2.3) where h n is defined as in (1.3) and the product is assumed to be one when m = 0 .Proof. The basic idea of our proof is to employ integration by parts and the inequalityin Lemma 2.1. By combining [3, Equation 18.9.7] and [3, Eqation 18.9.19], we have P n ( x ) = P n +1 ′ ( x ) − P n − ′ ( x )2 h − n , n ≥ , (2.4)3ubstituting this into (1.4) and applying integration by part once, we obtain a n = h − n Z − f ( x ) P n ( x ) dx = Z − f ( x ) P n +1 ′ ( x ) − P n − ′ ( x )2 dx = (cid:20) f ( x ) P n +1 ( x ) − P n − ( x )2 (cid:21) − + Z − f ′ ( x ) P n − ( x ) − P n +1 ( x )2 dx. (2.5)Furthermore, by making use of P n ( ±
1) = ( ± n for each n ≥
0, it is easy to see that thefirst term in the last equation vanishes and therefore a n = Z − f ′ ( x ) P n − ( x ) − P n +1 ( x )2 dx. (2.6)This together with the result of Lemma 2.1 gives | a n | ≤ Z − | f ′ ( x ) | " − x ) − p π (2 n −
1) + 2(1 − x ) − p π (2 n + 3) dx ≤ p π (2 n − Z − | f ′ ( x ) | (1 − x ) dx = 2 V p π (2 n − . This proves the case m = 0.When m = 1, integrating by part to (2.6) again, we get a n = Z − f ′ ( x )2 " P n ′ ( x ) − P n − ′ ( x )2 h − n − − P n +2 ′ ( x ) − P n ′ ( x )2 h − n +1 dx = " f ′ ( x ) P n ( x ) − P n − ( x )4 h − n − − − " f ′ ( x ) P n +2 ( x ) − P n ( x )4 h − n +1 − + Z − f ′′ ( x ) " P n − ( x )4 h − n − − P n ( x )4 h − n − − P n ( x )4 h − n +1 + P n +2 ( x )4 h − n +1 dx. (2.7)We see that the first two terms in the last equation vanish and therefore a n = Z − f ′′ ( x ) " P n − ( x )4 h − n − − P n ( x )4 h − n − − P n ( x )4 h − n +1 + P n +2 ( x )4 h − n +1 dx. (2.8)4y using the inequality in Lemma 2.1 again, we obtain | a n | ≤ Z − | f ′′ ( x ) | (1 − x ) " h n − p π (2 n −
3) + h n − p π (2 n + 1)+ h n +1 p π (2 n + 1) + h n +1 p π (2 n + 5) dx ≤ h n − p π (2 n − Z − | f ′′ ( x ) | (1 − x ) dx = 2 V p π (2 n − h n − , (2.9)where we have used the property that h n is strictly decreasing with respect to n in thesecond step and this proves the case m = 1.When m ≥
2, we may continue the above process and this brings in higher derivativesof f and corresponding higher variations up to V m . Hence we can obtain the desiredresult.How sharp is Theorem 2.2? We consider the following example f ( x ) = | x − t | , (2.10)where t ∈ ( − , x = t . In this case, it is readily verified that m = 1 and V m = 2(1 − t ) − and therefore the result of Theorem 2.2 can be written explicitly as | a n | ≤ V p π (2 n − (cid:18) n − (cid:19) − = 4(1 − t ) − p π (2 n − (cid:18) n − (cid:19) − . (2.11)Let B ( n ) denote the bound on the right-hand side of (2.11). We compare B ( n ) withthe absolute values of the Legendre coefficients | a n | and numerical results are illustratedin Figure 2 for two values of t . We can see clearly that, in the case of t = 0, i.e., f ( x ) = | x | , our bound B ( n ) is almost indistinguishable with | a n | as n increases. In fact,when n = 400, we have B ( n ) ≈ . | a n | ≈ . | a n | ≤ b V m ( n − )( n − ) · · · ( n − m + ) r π n − m − , (2.12)where n ≥ m + 2 and b V m = Z − | f ( m +1) ( x ) | (1 − x ) dx.
5e now make a comparison between the result of Theorem 2.2 with the bound on theright-hand side of (2.12). For simplicity, we let B ( n ) denote the bound on the right-handside of (2.12). For the function (2.10), we have b V = 2(1 − t ) − and thus B ( n ) = b V n − r π n −
2) = 2(1 − t ) − n − r π n − . (2.13)Comparing B ( n ) and B ( n ), we have for n ≥ B ( n ) B ( n ) = 2(1 − t ) π r n − n − < − t ) π . Clearly, we see that the new bound is always sharper; see Figure 2.Figure 2: The bound B ( n ) (line), the bound B ( n ) (dash) and | a n | (dots) for t = 0(left) and t = (right). Here n ranges from 5 to 400.We now consider the Legendre polynomial approximation by truncating the first N terms of (1.4), i.e., f N ( x ) = N − X n =0 a n P n ( x ) . (2.14)The following theorem is a corollary of Theorem 2.2. Theorem 2.3.
Under the assumptions of Theorem 2.2 and assume that m ≥ . • When m = 1 , then for each N ≥ , k f ( x ) − f N ( x ) k ∞ ≤ V p π (2 N − . (2.15)6 When m ≥ , then for each N ≥ m + 1 , k f ( x ) − f N ( x ) k ∞ ≤ V m ( m − p π (2 N − m − m Y k =2 h N − k . (2.16) Proof.
Recall the well-known inequality | P n ( x ) | ≤ x ∈ [ − , k f ( x ) − f N ( x ) k ∞ ≤ ∞ X n = N | a n | . (2.17)We first consider the case m = 1. By using Theorem 2.2, we obtain k f ( x ) − f N ( x ) k ∞ ≤ ∞ X n = N V m h n − p π (2 n −
3) = 2 V m √ π ∞ X n = N n − ) q n − . (2.18)Note that ∞ X n = N n − ) q n − ≤ ∞ X n = N n − ) ≤ Z ∞ N − (cid:18) x − (cid:19) − dx = 2 q N − . Substituting this into (2.18) gives the desired result.Next, we consider the case m ≥
2. Combining (2.17) and Theorem 2.2, we obtain k f ( x ) − f N ( x ) k ∞ ≤ ∞ X n = N V m p π (2 n − m − m Y k =1 h n − k ≤ V m p π (2 N − m − ∞ X n = N m Y k =1 h n − k . (2.19)Observe that m Y k =1 h n − k = 1 m − " m Y k =2 h n − k − m − Y k =1 h n − k , which implies ∞ X n = N m Y k =1 h n − k = 1 m − ∞ X n = N " m Y k =2 h n − k − m − Y k =1 h n − k = 1 m − m Y k =2 h N − k . (2.20)Substituting (2.20) into (2.19) gives the desired result. This completes the proof.7 emark . Note that the assumption in [5, Theorem 2.5] requires m >
1. Here wehave proved a result for the case m = 1.An interesting question is: What is the error bound of Legendre approximation to thefunction f ( x ) = | x | ? Note that the analysis of Chebyshev polynomial approximationsto this function has been discussed comprehensively in [4, Chapter 7]. Here we providea corresponding result for the truncated Legendre series. Corollary 2.5.
Let f ( x ) = | x | and let f N ( x ) be the truncated Legendre series of f ( x ) .Then, for each N ≥ , k f ( x ) − f N ( x ) k ∞ ≤ p π (2 N − . (2.21) Proof.
Note that m = 1 and V = 2 for this function. The bound follows immediatelyfrom Theorem 2.3. Remark . The result in Corollary 2.5 is actually overestimated. In fact, numericalexperiments show that k f ( x ) − f N ( x ) k ∞ = O ( N − ) as N → ∞ . However, a rigorousproof is still open. In this paper, we have presented a new and sharper bound for the Legendre coefficients ofdifferentiable functions. An illustrative example is provided to demonstrate the sharp-ness of our results. We further apply this result to obtain a new error bound of thetruncated Legendre series in the uniform norm.Finally, we remark that it is possible to extend the result of Theorem 2.2 to a moregeneral case. Indeed, from [3, Equation 18.14.7] we see that the Gegenbauer polynomialalso satisfies a Bernstein-type inequality, e.g.,(1 − x ) λ | C λn ( x ) | < − λ Γ( λ ) ( n + λ ) λ − , n ≥ , where x ∈ [ − ,
1] and 0 < λ < C λn ( x ) denotes the Gegenbauer polynomial ofdegree n . Therefore, one can expect that a sharp bound for the Gegenbauer coefficientsof differentiable functions can be obtained in a similar way. Acknowledgement
This work was supported by the National Natural Science Foundation of China undergrant 11671160. 8 eferences [1] V. A. Antonov and K. V. Holsevnikov, An estimate of the remainder in the expan-sion of the generating function for the Legendre polynomials (Generalization andimprovement of Bernstein’s inequality),
Vestnik Leningrad University Mathematics ,13, 163–166, 1981.[2] L. Lorch, Alternative proof of a sharpened form of Bernstein’s inequality for Leg-endre polynomials,
Applicable Analysis , 14, 237–240, 1983.[3] F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark,
NIST Handbook ofMathematical Functions , Cambridge University Press, 2010.[4] L. N. Trefethen,
Approximation Theory and Approximation Practice , SIAM, 2013.[5] H. Y. Wang and S. H. Xiang, On the convergence rates of Legendre approximation,
Mathematics of Computation , 81(278), 861–877, 2012.[6] H. Y. Wang, On the optimal estimates and comparison of Gegenbauer expansioncoefficients,
SIAM Journal on Numerical Aanlysis , 54(3), 1557–1581, 2016.[7] S. H. Xiang, On error bounds for orthogonal polynomial expansions and Gauss-typequadrature,
SIAM Journal on Numerical Analysis , 50(3), 1240–1263, 2012.[8] X. D. Zhao, L. L. Wang and Z. Q. Xie, Sharp error bounds for Jacobi expansions andGegenbauer–Gauss quadrature of analytic functions,